SIMULTANEOUS ESTIMATION OF SCALE MATRICES IN TWO …konno/pdf/tr1.pdf · 2010-03-01 ·...

J. Jpn. Soc. Comp. Statist., xx(200x), 1–32

SIMULTANEOUS ESTIMATION OF SCALE MATRICES IN

TWO-SAMPLE PROBLEM UNDER ELLIPTICALLY

CONTOURED DISTRIBUTIONS

Hisayuki Tsukuma∗ and Yoshihiko Konno†

Number of pages: Text (18), References (2), Tables (10), Figures (0), Total (32)

Abbreviated title: Estimating scale matrices under elliptically contuored distributions

Corresponding author: Hisayuki Tsukuma,

E-mail: [email protected]

∗Graduate School of Science and Technology, Chiba University, 1-33 Yayoi-cho, Chiba 263-8522, Japan


†Graduate School of Science and Technology, Chiba University, 1-33 Yayoi-cho, Chiba 263-8522, Japan


Key words: covariance matrix, Stein’s loss, Stein-Haff identity, two-sample problem

J. Jpn. Soc. Comput. Statist., Vol. xx, 200x

ABSTRACT

Two-sample problems of estimating p × p scale matrices are investigated under

elliptically contoured distributions. Two loss functions are employed; one is sum of

Stein’s loss functions of one-sample problem of estimating a normal covariance matrix

and the other is a quadratic loss function for Σ2Σ−11 , where Σ1 and Σ2 are p × p scale

matrices of elliptically contoured distribution models. It is shown that improvement

of the estimators obtained under the normality assumption remains robust under el-

liptically contoured distribution models. A Monte Carlo study is also conducted to

evaluate the risk performances of the improved estimators under three elliptically con-

toured distributions.

–2–

Compiled on April 26, 2002

1. Introduction

Since the pioneer works of Stein (1956) and James and Stein (1961), there has been a

great deal of effort to construct improved estimators for a covariance matrix of a multivari-

ate normal distribution. The literature includes Haff (1980, 1982, 1991) and Dey and Srini-

vasan (1985). Two sample analogue of estimating covariance matrices has been also consid-

ered by several authors such as Muirhead and Verathaworn (1985) and Loh (1991a, 1991b).

On the other hand Kubokawa and Srivastava (1999) showed that improvement of minimax

estimators for a covariance matrix obtained under the normality assumption remains robust

under elliptically contoured distribution models. In this paper, following the set-up consid-

ered by Loh (1991a, 1991b), we examine two-sample problems of estimating scale matrices

of elliptically contoured distributions.

The precise set-up of the problems is as follows: Let Y 1 and Y 2 be N1 × p and N2 × p

random matrices which take multivariate linear models of the form

Y 1 = C1β1 + ε1 and Y 2 = C2β2 + ε2. (1)

Here εi (i = 1, 2) are Ni ×p random matrices, Ci are known Ni ×m matrices with full rank,

and βi are unknown m × p matrices. We also assume that the error matrices ε1 and ε2

are marginally distributed as elliptically contoured distributions. But we assume the two

forms of the joint density functions of error matrices: First, two error matrices ε1 and ε2

are independently distributed and have the joint density function

Π2i=1|Σi|−Ni/2gi(tr(Σ−1

i ε′iεi)), (2)

where Σi (i = 1, 2) are p × p unknown positive definite matrices and gi are nonnegative

real-valued functions: Secondly two error matrices ε1 and ε2 are uncorrelatedly distributed

and have the joint density function

|Σ1|−N1/2|Σ2|−N2/2g(tr(Σ−11 ε′1ε1 + Σ−1

2 ε′2ε2)), (3)

where g is nonnegative real-valued function. Here |P |, tr(P ) and P ′ stand for the determi-

nant, the trace and the transpose of a square matrix P , respectively.

Following the approaches due to Loh (1991a, 1991b), we consider two sets of estimation

problems as follows.

–3–


(i) Under the model (1) with the assumption (2), the problem of estimating (Σ1, Σ2) with

unknown parameters (β1, β2) is considered under a loss function

L1(Σ1, Σ2, Σ1, Σ2) =2∑

i=1

{tr(ΣiΣ−1i ) − log |ΣiΣ

−1i | − p}, (4)

where Σi, i = 1, 2, are estimators of Σi, respectively. This loss function is a natural exten-

sion of Stein’s loss function in the one-sample case.

(ii) Under the model (1) with the assumption (3), the problem of estimating ζ = Σ2Σ−11

with unknown parameters (β1, β2) is considered under a loss function

L2(ζ, ζ) = tr{Σ−12 (ζ − ζ)S1(ζ − ζ)′}/trζ, (5)

where ζ is an estimator of ζ and S1 = Y ′1(IN1 − C1(C ′

1C1)−1C ′1)Y 1. This estimation

problem is related to estimation of the common mean of two multivariate distributions. See

a possible motivation for Loh (1991b). Furthermore, the eigenvalues of ζ are important, for

example, in the problem of testing the null hypotheses Σ1 = Σ2 against the alternative hy-

potheses Σ1 �= Σ2. For estimating these eigenvalues, see Muirhead and Verathaworn (1985),

Muirhead (1987), and DasGupta (1989).

This paper is organized in the following way. In Section 2, we treat the problem (i). We

adapt the extended Stein and Haff identity due to Kubokawa and Srivastava (1999) for two

sample set-up (which is stated in Section 4) and obtain sufficient conditions under which an

alternative estimator improves upon the James-Stein estimator (T 1D1T′1, T 2D2T

′2) with

respect to the loss function (4). Here T i, i = 1, 2, is the lower triangular matrix with positive

diagonal elements such that Si = T iT′i and Di is diagonal matrix with the j-th diagonal ele-

ment 1/(Ni−m+p+1−2j), j = 1, 2, . . . , p, where Si = Y ′i(INi −Ci(C ′

iCi)−1C ′i)Y i, i =

1, 2. Simulation study is conducted to evaluate risk performances of alternative estima-

tors under the multivariate normal distribution, the matrix-variate t-distribution, and the

matrix-variate Kotz-type distribution. Since these matrix-variate distributions except the

normal distribution are not independent sampling, we also conduct simulation study based

on independently and identically sampling model from the multivariate t-distribution and the

Kotz-type distribution, respectively. Finding in this Section is that the estimators obtained

under the error distribution (2) (i.e., which is different from independently and identically

sampling model) perform well under independently and identically sampling from the ellip-

tically contoured error models. In Section 3, we treat the problem (ii). In this problem, we

–4–


treat the joint density function (3) only since we fail to obtain the suitable integration-by-

parts formula under the joint density (2) to get improved estimators. We first obtain the

best estimator among the constant multiple of S2S−11 . Next we consider several types of

improvement over the best constant multiple of S2S−11 and conduct simulation study in the

much same way as that of Section 2. The proofs of the results obtained in Sections 2 and 3

put into Section 4.

2. Simultaneous estimation of (Σ1,Σ2)

To consider the estimation problem of (Σ1,Σ2), we shall employ the loss function (4)

and evaluate performance of estimators of (Σ1, Σ2) by means of their risk function, i.e.,

E[L1(Σ1, Σ2, Σ1, Σ2)], where the expectation is taken with respect to the joint distribu-

tions of two error distributions (2).

2.1. Class of estimators

First we introduce estimators obtained from one-sample problem of estimating a normal

covariance matrix. We define the usual estimator

(ΣUS

1 , ΣUS

2 ) = (S1/n1,S2/n2), (6)

where ni = Ni −m, i = 1, 2. Also we put the James-Stein estimator

(ΣJS

1 , ΣJS

2 ) = (T 1D1T′1, T 2D2T

′2), (7)

where T i, i = 1, 2, is the lower triangular matrix with positive diagonal elements such that

Si = T iT′i and Di is diagonal matrix with the j-th diagonal element 1/(ni + p + 1 −

2j), j = 1, 2, . . . , p. Note that the James-Stein estimator (7) is invariant under the group

of transformations given by

Σi → P iΣiP′i, Si → P iSiP

′i, i = 1, 2,

where P i is any p × p lower triangular matrix with positive diagonal elements. ¿From

the argument from James and Stein (1961), we can see that the estimator (7) has smaller

risk than that of the estimator (6). Furthermore, applying Proposition 1 in Kubokawa and

Srivastava (1999), we can see that improvement of the estimator (7) over the usual estimator

(6) remains robust for all possible functions g1 and g2 in (2). To improve upon the estimator

–5–


(7) by using both S1 and S2 simultaneously, we adapt argument due to Loh (1991) and

consider a class of invariant estimator under the group of the transformations

Σi → QΣiQ′, Si → QSiQ

′, i = 1, 2, (8)

where Q is any p× p nonsingular matrix. From Loh (1991a), we can see that an invariant

estimator under the above group transformations has the form

(ΣEQ

1 , ΣEQ

2 ) = (B−1Ψ(F )B′−1, B−1Φ(F )B′−1). (9)

Here we assume that B is a nonsingular matrix such that B(S1 + S2)B′ = Ip, BS2B′ =

F , and F = diag(f1, f2, . . . , fp) with f1 ≥ f2 ≥ · · · ≥ fp > 0, and that Ψ(F ) =

diag(ψ1(F ), ψ2(F ), . . . , ψp(F )) and Φ(F ) = diag(φ1(F ), φ2(F ), , . . . , φp(F )) are diago-

nal matrices whose elements are functions of F . In the sequel of the paper, we abbreviate

Ψ, Φ, ψi, φi (i = 1, 2, . . . , p) for Ψ(F ), Φ(F ), ψi(F ), φi(F ), respectively.

2.2. A sufficient conditions for improvement upon the James-Stein estimator

Now we state the main result in this section.

Theorem 1 Suppose that we wish to estimate (Σ1, Σ2) simultaneously under the loss func-

tion (4). An invariant estimator (ΣEQ

1 , ΣEQ

2 ) is better than the James-Stein estimator

(ΣJS

1 , ΣJS

2 ) for arbitrary g1 and g2 in (2) if

(i) p− (n1 − p− 1)p∑

j=1

ψj

1 − fj− 2

p∑j=1

[ψj + fj

∂ψj

∂(1 − fj)+ ψj

∑k �=j

fk

fk − fj

]≥ 0,

(ii) p− (n2 − p− 1)p∑

j=1

φj

fj− 2

p∑j=1

[φj + (1 − fj)

∂φj

∂fj+ φj

∑k �=j

1 − fk

fj − fk

]≥ 0,

(iii)p∑

j=1

[−(log d1j + log d2j) + log

ψj

1 − fj+ log

φj

fj

]≥ 0,

where d1j = 1/(n1 + p+ 1 − 2j) and d2j = 1/(n2 + p+ 1 − 2j), j = 1, 2, . . . , p.

As a special case of (ΣEQ

1 , ΣEQ

2 ), we introduce the estimator due to Loh (1991)

(ΣLO

1 , ΣLO

2 ) = (B−1ΨLOB′−1, B−1ΦLOB′−1), (10)

where ΨLO = diag(ψLO1 , . . . , ψLO

p ) and ΦLO = diag(φLO1 , . . . , φLO

p ) with the j-th diagonal

elements ψLOj = (1− fj)/(n1 − p− 1 + 2j) and φLO

j = fj/(n2 + p+ 1− 2j), j = 1, 2, . . . , p,

respectively.

–6–


Immediately we get the following corollary from Theorem 1.

Corollary 1 The estimator (ΣLO

1 , ΣLO

2 ) is better than (ΣJS

1 , ΣJS

2 ) for arbitrary g1 and g2

in (2).

Remark : When the error (ε1, ε2) have the joint density (3), we can see the dominance

results similar to those in Theorem and Corollary above.

2.3. Numerical studies

Furthermore we introduce the Dey-Srinivasan estimator

(ΣDS

1 , ΣDS

2 ) = (H1D1K1H′1, H2D2K2H

′2), (11)

where Hi, i = 1, 2, is a p × p orthogonal matrix such that Si = HiKiH′i and Ki =

diag(ki1, ki2, . . . , kip) with ki1 ≥ ki2 ≥ · · · ≥ kip > 0.

Note that the Dey-Srinivasan estimator (11) is invariant under the group of transfor-

mations given by

Σi → OiΣiO′i, Si → OiSiO

′i, i = 1, 2,

where Oi is any p × p orthogonal matrix. Furthermore from Proposition 1 in Kubokawa

and Srivastava (1999), we can see that the estimator (11) improves upon the estimator (7).

However, it is difficult to compare (ΣLO

1 , ΣLO

2 ) with (ΣDS

1 , ΣDS

2 ) analytically. Therefore,

to compare the risk performances of these estimators, we carry out Monte Carlo simulations.

Our simulations are based on 10,000 independent replications. We consider three-type

of error distributions which are given in the following.

1. The matrix-variate normal distribution: The joint density function (ε1, ε2) is given by

Π2i=1ci1|Σi|−Ni/2 exp[−(1/2)tr(Σ−1

i ε′iεi)],

where ci1 = (2π)−Nip/2.

2. The t-distribution: The joint density function (ε1, ε2) is given by

Π2i=1ci2|Σi|−Ni/2{1 + (1/vi)tr(Σ−1

i ε′iεi)}−(vi+Nip)/2,

where ci2 = Γ[{vi +Nip}/2]/{(πvi)Nip/2Γ[vi/2]}, vi > 0. Here we denote by Γ( · ) the

Gamma function.

–7–


3. The Kotz-type distribution: The joint density function (ε1, ε2) is given by

Π2i=1ci3|Σi|−Ni/2{tr(Σ−1

i ε′iεi)}ui−1 exp[−ri{trΣ−1i ε′iεi}si],

where ri > 0, si > 0, 2ui +Nip > 2, and

ci3 =siΓ[Nip/2]r{ui+Nip/2−1}/si

i

πNip/2Γ[{ui +Nip/2 − 1}/si].

For generating a random number of the Kotz-type distribution above, see Fang, Kotz,

and Ng (1990) for example.

For Monte Carlo simulations, we took N1 = N2 = 15, m = 1, and p = 3 and we also

put v1 = v2 = 3 for the t-distribution and (ui, ri, si) = (5, 0.1, 2), i = 1, 2, for the Kotz-type

distribution. We also suppose that β1 = β2 = (0, 0, 0)′ and that the parameter Σ2Σ−11 is

the diagonal matrix with typical elements. The estimated risks of these cases are given by

Tables 1–3 and their estimated standard errors are in parentheses.

In Tables 1–3, ‘US’, ‘JS’, ‘DS’, and ‘LO’ stand for the usual estimator (ΣUS

1 , ΣUS

2 ),

the James-Stein estimator (ΣJS

1 , ΣJS

2 ), the Dey-Srinivasan estimator (ΣDS

1 , ΣDS

2 ), and the

Loh estimator (ΣLO

1 , ΣLO

2 ), respectively and ‘AI’ stands for average of improvement in risk

over (ΣUS

1 , ΣUS

2 ).

We also carried out simulations when the rows of εi(i = 1, 2) have densities

|Σi|−Ni/2h(e′ijΣ

−1i eij), for j = 1, . . . , Ni,

where εi = (e′i1, e′

i2, . . . , e′iNi

)′. That is, the rows of each error matrix εi are independently

and identically distributed (i.i.d.) as an elliptically contoured distribution.

For Monte Carlo simulations, we suppose that the rows of εi follow the vector-valued t-

distributions, i.e., the density function of the random vectors eij (i = 1, 2, j = 1, 2, . . . , Ni)

are given by

ci4 |Σi|−1/2(1 + e′ijΣ

−1i eij/vi)−(vi+p)/2, (12)

where vi > 0 and ci4 = Γ[(vi +p)/2]/{(πvi)p/2Γ[vi/2]}, and we also suppose that the rows of

εi follow the vector-valued Kotz-type distributions, i.e., the density functions of the random

vectors eij (i = 1, 2, j = 1, 2, . . . , Ni) are given by

ci5|Σi|−1/2{e′ijΣ

−1i eij}li−1 exp[−ri{e′

ijΣ−1i eij}si], (13)

–8–


where ri > 0, si > 0, 2ui + p > 2, and

ci5 =siΓ[p/2]r{ui+p/2−1}/si

i

πp/2Γ[{ui + p/2 − 1}/si].

For simulations, we take N1 = N2 = 15, m = 1, and p = 3 and we put v1 = v2 = 3 for

the t-distributions and (ui, ri, si) = (5, 0.1, 2), i = 1, 2, for the Kotz-type distributions. The

estimated risks of these cases are given by Tables 4 and 5.

The results of Monte Carlo simulations indicate that

1. When the eigenvalues of Σ2Σ−11 are close together, AI of LO is large;

2. DS is better than LO;

3. AIs of all alternative estimators are relatively small under non-normal error;

4. AIs of LO and DS are substantial under independently and identically sampling set-up

from non-normal distribution, although we cannot prove improvement of alternative

estimators over the usual estimator under this situation. Hence, these results suggest

that the improvement under densities (2) remains robust even if the rows of errors are

i.i.d.

3. Estimation of Σ2Σ−11

In this section we consider the problem (ii) given in Section 1 under elliptical error with

density (3) and we treat the problem under the loss function

L2(ζ; ζ) = tr{Σ−12 (ζ − ζ)S1(ζ − ζ)′}/trζ, (14)

as considered by Loh (1988). Recall that Si = Y ′i(INi −Ci(C ′

iCi)−1C ′i)Y i for i = 1, 2. As

pointed out in Loh (1991b), the problem (ii) is invariant under the group of transformations

given by (8) and the estimators which is invariant under this transformation group has the

form

ζ = A−1ΞA,

where A is a nonsingular matrix such that AS1A′ = Ip and AS2A

′ = L with L =

diag(l1, . . . , lp) (l1 ≥ l2 ≥ · · · ≥ lp > 0) and further Ξ = diag(ξ1, . . . , ξp) whose diagonal

elements are functions of L.

–9–


3.1. The best constant multiplier of S2S−11

Consider a class of estimators of the form ζUS

= αS2S−11 , where α is a constant. Then

this estimator can be rewritten as

ζUS

= A−1ΞUSA,

where ΞUS is diagonal matrix whose the j-th diagonal element is ξUSj = αlj, j = 1, 2, . . . , p.

Theorem 2 For any function g in (3), the best usual estimator of ζ under the loss function

(14) is given by

ζBU

= A−1ΞBUA, (15)

where Ξ = diag(ξBU1 , . . . , ξBU

p ) with ξBUj = [(n1 − p− 1)/(n2 + p+ 1)]lj .

3.2. Improved Estimators

We next discuss an improvement on the estimator (15). It is expected that the eigen-

values of S2S−11 are more spread out than those of Σ2Σ−1

1 . To reduce the biases of the

estimators for eigenvalues, we consider

ΞLO = diag(ξLO1 , . . . , ξLO

p ), ξLOj = (n1 − p− 1)lj/(n2 + p + 3 − 2j), (16)

for j = 1, 2, . . . , p. Then we have the following theorem:

Theorem 3 Under the loss function (14), ζLO

= A−1ΞLOA is better than ζBU

for any

function g in (3).

Further we consider an improved estimator on ζLO as in Loh (1988). Define the Berger-

type estimator as

ζBE

= A−1ΞBEA, (17)

where ΞBE = diag(ξBE1 , . . . , ξBE

p ) with

ξBEj = ξLO

j +c

b+ u, u =

p∑j=1

(n2 + p+ 3 − 2j(n1 − p− 1)lj

)2

.

Here c : R+ �→ R is a differentiable function of u and b is a suitable positive constant.

Then we have the following theorem:

Theorem 4 Assume that

–10–


(I) p ≥ 3, n1 ≥ p and n2 ≥ p;

(II) c(u) ≥ 0 and c′(u) ≥ 0 for all u ≥ 0;

(III) supu c(u)/√b ≤ 4(p2 + p− 4)(n2 − p+ 3)/[√p(n1 − p− 1)(n2 − p+ 7)].

Then, under the loss function (14), ζBE

is better than ζLO

for any function g in (3).

Since l1 ≥ · · · ≥ lp, the diagonal elements of Ξ should have the ordering property, i.e.,

ξ1 ≥ · · · ≥ ξp. Hence, to improve on ζLO

, we can consider the estimator, for example, with

Stein’s isotonic regression on the ξLOj ’s (see Lin and Perlman, 1985).

Further we can also consider the Stein-type estimator (Stein, 1977)

ζST

= A−1ΞST A, (18)

where ΞST = diag(ξST1 , . . . , ξST

p ) with

ξSTj = (n1 − p− 1)lj

/(n2 + p+ 1 + 2

∑k �=j

lklj − lk

), j = 1, 2, . . . , p.

The derivation of this estimator is given by Loh (1991b).

By applying methods of Berger and of Stein, the Stein-Berger estimator is given by

ζSB

= A−1ΞSBA,

where ΞSB = diag(ξSB1 , . . . , ξSB

p ) with

ξSBj = ξST

j +c

b+ u(j = 1, 2, . . . , p), u =

p∑j=1

(n2 + p+ 3 − 2j(n1 − p− 1)lj

)2

,

and the ξSTj ’s are constructed by Stein’s isotonic regression on the ξST

j ’s. However, we

cannot analytically compare ζBE

with ζSB

and hence, in the next subsection, we examine

these risk performances by using a numerical study.

3.3. Numerical studies

We have carried out Monte Carlo simulations (10,000 runs) to observe the risk perfor-

mances of several estimators in the previous subsection.

1. The matrix-variate normal distribution: The joint density function of (ε1, ε2) is given

by

γ1|Σ1|−N1/2|Σ2|−N2/2 exp[−(1/2)tr{Σ−11 ε′1ε1 + Σ−1

2 ε′2ε2}],where γ1 = (2π)−(N1+N2)p/2.

–11–


2. The t-distribution: The joint density function of (ε1, ε2) is given by

γ2 |Σ1|−N1/2|Σ2|−N2/2[1 + (1/v)tr{Σ−11 ε′1ε1 + Σ−1

2 ε′2ε2}]{−(v+(N1+N2)p}/2,

where γ2 = Γ[{v + (N1 +N2)p}/2]/{(πv)(N1+N2)p/2Γ[v/2]}, v > 0.

3. The Kotz-type distribution: The joint density function of (ε1, ε2) is given by

γ3|Σ1|−N1/2|Σ2|−N2/2[tr{Σ−11 ε′1ε1 + Σ−1

2 ε′2ε2}]u−1

× exp[−rtr{Σ−11 ε′1ε1 + Σ−1

2 ε′2ε2}s],

where r > 0, s > 0, 2u+ (N1 +N2)p > 2, and

γ3 =sΓ[(N1 +N2)p/2]r{u+(N1+N2)p/2−1}/s

π(N1+N2)p/2Γ[{u+ (N1 +N2)p/2 − 1}/s] .

The estimated risks with the error above are given in Tables 6–8. For Monte Carlo

simulations, we took N1 = N2 = 15, m = 1, and p = 3 and we also put v = 5 for the

t-distribution and (u, r, s) = (5, 0.1, 2) for the Kotz-type distribution. We supposed that

β1 = β2 = (0, 0, 0)′ and that the parameter Σ2Σ−11 is the diagonal matrix with typical

elements.

In tables, ‘BU ’, ‘LO’, ‘BE’, and ‘SB’ stand for ζBU

, ζLO, ζ

BErespectively and ζ

SB,

and ‘AI’ stands for the average of improving over risk of ζBU

. For ζBE

and ζSB

, we set

b = 100 and

c =2(p2 + p − 4)(n2 − p+ 3)

√b√

p(n1 − p− 1)(n2 − p+ 7).

We also studied simulations when the rows of the error are independently and identically

distributed as in the densities (12) and (13). We assume that the distributions of rows of

the errors are (12) and are (13) in the previous section. This results are given in Tables 9

and 10. We got the similar results as those in Tables 7 and 8.

We summarize our numerical results in Tables 6–10 as follows:

1. When the diagonal elements of Σ2Σ−11 are close together, AIs of BE and SB are

substantially large under normal error as well as under non-normal error;

2. Among the alternative estimators, AI of SB is the largest when the diagonal elements

of Σ2Σ−11 are equal or when the smallest diagonal element of Σ2Σ−1

1 is far from the

others and the others are large.

–12–


3. When the largest diagonal element of Σ2Σ−11 is far from the second, AIs are relatively

small.

4. The results in Tables 9 and 10 suggest that the improvement by ζLO

and ζBE

under

density (3) remains robust even if the rows of errors are i.i.d.

4. Proofs of Theorems

In this section we show theorems and corollaries in Sections 2 and 3. To give the proofs,

we state a canonical form of our problems and list useful lemmas. Listed lemmas consist

of two ingredients. First, we adapted the integration-by-parts formula from Kubokawa and

Srivastava (1999) for our problems. We introduce two types of integration-by-parts formulas

which concerns the joint density function (2) and the joint density function (3), respectively.

Second, we quote lemmas on eigenstructure from Loh (1988).

4.1. Preliminaries

To derive a canonical form, write (ΓiCi)′ = ((C′iCi)1/2, 0), i = 1, 2, where Γi is an

Ni × Ni orthogonal matrix. Also put θi = (C ′iCi)1/2βi and ni = Ni −m. Furthermore,

write (X′i, Z′

i)′ = ΓiY i, where X i and Zi are m×p and ni ×p matrices respectively. Then

the densities (2) are rewritten as

|Σi|−Ni/2gi[tr{Σ−1i (X i − θi)′(X i − θi)} + tr(Σ−1

i Z ′iZi)] (19)

for i = 1, 2.

Next we introduce notation for integration-by-parts formula with respect to the joint

densities (2). Let

Gi(x) =12

∫ +∞

x

gi(t)dt

and let θ = (θ1, θ2) and Σ = (Σ1,Σ2). For a function U ≡ U(X1, X2, Z1, Z2), define

Eg1g2θ, Σ [U ] =

∫U ×

{ 2∏i=1

|Σi|−Ni/2gi(wi)}dX1dX2dZ1dZ2, (20)

EG1g2θ, Σ [U ] =

∫U × |Σ1|−N1/2|Σ2|−N2/2G1(w1)g2(w2)dX1dX2dZ1dZ2, (21)

Eg1G2θ, Σ [U ] =

∫U × |Σ1|−N1/2|Σ2|−N2/2g1(w1)G2(w2)dX1dX2dZ1dZ2, (22)

where wi = tr{Σ−1i (X i − θi)′(Xi − θi)} + tr(Σ−1

i Z ′iZi) (i = 1, 2).

–13–


Put Si = Z ′iZi and recall that Si = Y ′

i{INi − Ci(C ′iC)−1C ′

i}Y i for i = 1, 2 and let

H ≡ H(S1, S2) = (hij) be a p× p matrix such that the (j, k)-element hjk is a function of

S1 = (s1·jk) and S2 = (s2·jk). For i = 1, 2, let

{DSiH}jk =p∑

a=1

di·jahak, (23)

where

di·ja =12(1 + δja)

∂

∂si·ja

with δja = 1 for j = a and δja = 0 for j �= a. Also put Zi = (z′i1, . . . , z

′ini

)′ and zij =

(zi·j1, . . . , zi·jp) for i = 1, 2, and j = 1, 2, . . . , ni. Hence we have Si = Z ′iZi =

∑ni

j=1 z′ijzij

for i = 1, 2.

Now we adapt the extended Stein-Haff identity due to Kubokawa and Srivastava (1999)

for our problem. The difference between derivations of our identity and of that by Kubokawa

and Srivastava (1999) is what expectation for the variables of integration are multiplied.

Hence, we state the following formula without the proof.

Lemma 1 Let

Hi ≡ Hi

( n1∑j1=1

z′1j1

z1j1,

n2∑j2=1

z′2j2

z2j2

), i = 1, 2,

be a p× p matrix whose element is differentiable with respect to zi·jk (ji = 1, 2, . . . , ni, k =

1, 2, . . . , p). Furthermore, assume that

(a) Eg1g2θ, Σ

[∣∣trHiΣ−1i

∣∣] (i = 1, 2) is finite;

(b) limzi·jk→±∞ |zi·jk|H i

( n1∑j1=1

z′1j1

z1j1 ,

n2∑j2=1

z′2j2

z2j2

)( ni∑ji=1

z′1ji

z1ji

)−1

Gi(zi·jk + a) = 0

for any real a.

Then we have

Eg1g2θ, Σ [tr(H1Σ−1

1 )] + Eg1g2θ, Σ [tr(H2Σ−1

2 )] = EG1g2θ, Σ

[(n1 − p− 1)tr(H1S

−11 ) + 2tr(DS1H1)

]+Eg1G2

θ, Σ

[(n2 − p− 1)tr(H2S

−12 ) + 2tr(DS2H2)

]for θ = (θ1, θ2) and Σ = (Σ1,Σ2).

–14–


To derive the integration-by-parts formula with respect to the density (3), we make an

orthogonal transformation Y 1 and Y 2 to rewrite the density (3) as

|Σ1|−N1/2|Σ2|−N2/2g

{ 2∑i=1

[tr{Σ−1

i (X i − θi)′(Xi − θi)} + tr(Σ−1i Z ′

iZi)]}, (24)

where X i, Zi and θi are defined in the same way to obtain (19). For a real-valued function

U , denote

Egθ, Σ[U ] =

∫U ×

( 2∏i=1

|Σi|−Ni/2

)g(w)dX1dX2dZ1dZ2,

EGθ, Σ[U ] =

∫U ×

( 2∏i=1

|Σi|−Ni/2

)G(w)dX1dX2dZ1dZ2,

where w =∑2

i=1[tr{Σ−1i (Xi − θi)′(X i − θi)} + tr(Σ−1

i Z ′iZi)], G(x) = (1/2)

∫ +∞x g(t)dt,

θ = (θ1, θ2), and Σ = (Σ1, Σ2). ¿From preliminaries as above, we get the integration-by-

parts formula for the density (3):

Lemma 2 Assume that

H ≡ H(S1, S2) (25)

is differentiable with respect to zi·jk (i = 1, 2, j = 1, 2, . . . , ni, k = 1, 2, . . . , p) and that

(a) Egθ, Σ

[∣∣tr(HΣ−1i )

∣∣] is finite for i = 1, 2;

(b) limzi·jk→±∞ |zi·jk|H(S1,S2

)(Si)−1G(zi·jk + a) = 0 where Si =

∑ni

ji=1 z′iji

ziji, a is

any real number and i = 1, 2.

Then, for i = 1, 2,

Egθ, Σ[tr(HΣ−1

i )] = EGθ, Σ

[(ni − p− 1)tr(HS−1

i ) + 2tr(DSiH)]. (26)

Furthermore, we need the following lemmas to show main theorems and their corollaries.

Lemma 3 [Loh, 1991a] Let S1 and S2 be p × p symmetric and positive-definite matrices.

Also let B be nonsingular matrix such that B(S1 + S2)B′ = Ip, BS2B′ = F where F =

diag(f1, f2, . . . , fp) with f1 ≥ f2 ≥ · · · ≥ fp. Furthermore, let Ψ = diag(ψ1, ψ2, · · · , ψp)

and Φ = diag(φ1, φ2, · · · , φp), where the ψj and the φj (j = 1, 2, . . . , p) are differentiable

–15–


functions of F . Then we have

tr(DS1B−1ΨB′−1) =

p∑j=1

[ψj + fj

∂ψj

∂(1 − fj)+ ψj

∑k �=j

fk

fk − fj

],

tr(DS2B−1ΦB′−1) =

p∑j=1

[φj + (1 − fj)

∂φj

∂fj+ φj

∑k �=j

1 − fk

fj − fk

],

where DS1 and DS2 are defined as (23).

Lemma 4 [Loh, 1991b] Let S1 and S2 be p × p symmetric and positive-definite matri-

ces. Also let A be nonsingular matrix such that AS1A′ = Ip, AS2A

′ = L where L =

diag(l1, l2 . . . , lp) with l1 ≥ l2 ≥ · · · ≥ lp. Let Ξ = diag(ξ1, ξ2, . . . , ξp), where the

ξj (j = 1, 2, . . . , p) are differentiable functions of L. Then we have

tr(DS1A−1ΞA′−1) =

p∑j=1

[ξj − lj

∂ξj∂lj

+ ξj∑k �=j

lklk − lj

],

tr(DS2A−1Ξ2A′−1) =

p∑j=1

[2ξj

∂ξj∂lj

+ ξ2j∑k �=j

1lj − lk

],

where DS1 and DS2 are defined as (23).

4.2. Proofs of Theorem 1 and Corollary 1 in Section 2

Proof of Theorem 1: ¿From Lemmas 1 and 3, the risk of the estimator (Σ1, Σ2) can be

expressed as

R1(Σ1, Σ2) = EG1g2θ, Σ

[(n1 − p− 1)

p∑j=1

ψj

1 − fj+ 2

p∑j=1

{ψj + fj

∂ψj

∂(1 − fj)+ ψj

∑k �=j

fk

fk − fj

}]

+Eg1G2θ, Σ

[(n2 − p − 1)

p∑j=1

φj

fj+ 2

p∑j=1

{φj + (1 − fj)

∂φj

∂fj+ φj

∑k �=j

1 − fk

fj − fk

}]

+Eg1g2θ, Σ

[−

p∑j=1

{log

ψj

1 − fj+ log

φj

fj

}+

2∑i=1

{− log |Si|+ log |Σi| − p}]. (27)

Similarly the risk of the James-Stein estimator (ΣJS

1 , ΣJS

2 ) is given by

R1(ΣJS

1 , ΣJS

2 ) = EG1g2θ, Σ

[p]+ Eg1G2

θ, Σ

[p]

+Eg1g2θ, Σ

[−

p∑j=1

{log d1j + log d2j

}+

2∑i=1

{− log |Si| + log |Σi| − p}], (28)

–16–


where dij = 1/(ni + p + 1 − 2j). Hence, comparing the integrands with respect to each

expectation of (20), (21), and (22) in the rhs of the equations (27) and (28), we complete

the proof. �

Proof of Corollary 1: ¿From Theorem 1, it suffices to show that

p − (n1 − p + 1)p∑

j=1

d∗1j − 2p∑

j=1

[(1 − fj)d∗1j

∑k �=j

fk

fk − fj

]≥ 0, (29)

p− (n2 − p+ 1)p∑

j=1

d∗2j − 2p∑

j=1

[fjd

∗2j

∑k �=j

1 − fk

fj − fk

]≥ 0, (30)

where d∗1j = 1/(n1 − p− 1 + 2j) and d∗2j = 1/(n2 + p + 1 − 2j). We here note that the last

term of the lhs in (29) is evaluated as

−p∑

j=1

{(1 − fj)d∗1j

∑k �=j

fk

fk − fj

}=

∑j<k

fk(1 − fj)d∗1j − fj(1 − fk)d∗1k

fj − fk

=∑j<k

fk(1 − fj)(d∗1j − d∗1k)fj − fk

−∑j<k

d∗1k

≥ −∑j<k

d∗1k,

where the last inequality in the above display is derived by the fact that fk < fj < 1 and

d∗1k < d∗1j for j < k. Since∑

j<k d∗1k =

∑pj=1(j − 1)d∗1j , we get

p− (n1 − p+ 1)p∑

j=1

d∗1j − 2p∑

j=1

{(1 − fj)d∗1j

∑k �=j

fk

fj − fk

}

≥ p−p∑

j=1

(n1 − p− 1 + 2j)d∗1j = 0.

The proof of the inequality (30) can proceed similarly. Note that the last term of the lhs in

(30) is evaluated as

−p∑

j=1

[fjd

∗2j

∑k �=j

1 − fk

fj − fk

]=

∑j>k

fk(1 − fj)(d∗2k − d∗2j)fj − fk

−∑j>k

d∗2k ≥ −p∑

j=1

(p− j)d∗2j .

Putting the above inequality into the lhs of (30), we get the desired result. �

4.3. Proofs of Theorems 2, 3, and 4 in Section 3

First, we give the following lemma:

–17–


Lemma 5 For estimation of ζ = Σ2Σ−11 in model (3), we consider an estimator of the

form

ζ = A−1ΞA, (31)

where A is a nonsingular matrix such that AS1A′ = Ip and AS2A

′ = L with L =

diag(l1, l2, . . . , lp) with l1 ≥ l2 ≥ · · · ≥ lp > 0, and Ξ = diag(ξ1, ξ2, . . . , ξp). Then,

under the loss function (5), the risk of ζ is given by

R2(ζ; ζ) = EGθ, Σ

[n1 +

∑j

{n2 − p− 1

ljξ2j + 2ξ2j

∑k �=j

1lj − lk

+ 4ξj∂ξj∂lj

−2(n1 − p+ 1)ξj + 4ξj∑k �=j

lklj − lk

+ 4lj∂ξj∂lj

}/trζ

]. (32)

Proof. We can write the risk function as

R2(ζ; ζ) = Egθ, Σ[tr{Σ−1

1 Σ2Σ−11 S1 + Σ−1

2 A−1Φ2A′−1 − 2Σ−11 A−1ΦA′−1}/ trζ].

Noting that Egθ, Σ[S1] = EG

θ, Σ[n1Σ1] and using Lemmas 2 and 4, we get (32). �

Proof of Theorem 2: Substituting αlj for ξj in (32), we get

R2(ζUS

; ζ) = EGθ, Σ[n1 + {(n2 + p+ 1)α2 − 2(n1 − p− 1)α}

p∑j=1

lj/

trζ]. (33)

Hence we can see that (33) is minimized at α = (n1 − p− 1)/(n2 + p+ 1). �

Proof of Theorems 3: The proof proceed in similar way as in that of Theorem 3.5 in Loh

(1988). We reproduce it for reader’s convenience. Write dj = (n1−p−1)/(n2+p+3−2j), j =

1, 2, . . . , p. ¿From (33), we have

(trζ){R2(ζBU

; ζ) − EGθ, Σ[n1]} = EG

θ, Σ

[−(n1 − p− 1)2

n2 + p+ 1

p∑i=1

li

].

¿From Lemma 5, we have

(trζ){R2(ζLO

; ζ) − EGθ, Σ[n1]} = EG

θ, Σ

[p∑

j=1

{(n2 − p+ 3)d2

j lj + 2∑k<j

d2k(l2j − l2k)lj − lk

+2∑k<j

l2j (d2j − d2

k)lj − lk

− 2(n1 − p− 1)djlj + 4∑k<j

ljlk(dj − dk)lj − lk

}]. (34)

–18–


Noting that

2∑k<j

l2j (d2j − d2

k)lj − lk

+ 4∑k<j

lj lk(dj − dk)lj − lk

≤ 0

and thatp∑

j=1

∑k<j

d2k(l2j − l2k)lj − lk

=p∑

j=1

{(p− j)d2

j +∑k<j

d2k

}lj ,

we can see that the rhs of (34) is less than

EGθ, Σ

p∑j=1

{(n2 + p+ 3 − 2j)d2

j − 2(n1 − p− 1)dj + 2∑k<j

d2k

}lj

= EG

θ, Σ

[p∑

j=1

{−(n1 − p− 1)2

n2 + p + 1− 2(n1 − p− 1)2

(j − 1

(n2 + p+ 1)(n2 + p+ 3 − 2j)

−∑k<j

1(n2 + p+ 3 − 2k)2

)}lj

].

Furthermore, from mathematical induction on j, we can see that

j − 1(n2 + p+ 1)(n2 + p+ 3 − 2j)

≥∑k<j

1(n2 + p+ 3 − 2k)2

.

¿From this inequality, we finally get

(trζ){R2(ζLO

; ζ) − EGθ, Σ[n1]} ≤ (trζ){R2(ζ

BU; ζ) − EG

θ, Σ[n1]},

which completes the proof. �

Proof of Theorem 4: The proof proceed in the same way as in that of Theorem 3.5 in

Loh (1988). However, we reproduce it for reader’s convenience when c is a positive constant

which satisfies Assumption (III). Put αj = c/{djlj(b + u)}, j = 1, 2, . . . , p. Hence ξBEj =

djlj(1 + αj). Also note that

∂ξBEj

∂lj= dj(1 + αj) + djlj

∂αj

∂lj= dj(1 + αj) +

c

lj(b+ u)

[2

d2j l

2j (b + u)

− 1].

¿From tedious calculation, we have

(trζ){R2(ζBE

; ζ) − R2(ζLO

; ζ)}

= EGθ, Σ

p∑j=1

[(n2 − p− 1)d2

j lj(2αj + α2j) + 4

∑k<j

d2j l

2jαj − d2

kl2kαk

lj − lk+ 4d2

j lj(2αj + α2j )

–19–


+4d2j l

2j (1 + αj)

∂αj

∂lj− 2(n1 − p+ 1)djljαj + 4

∑k<j

dj ljαjlk − dklkαkljlj − lk

+4lj djαj + 4djl2j

∂αj

∂lj

]= EG

θ, Σ

p∑j=1

[(n2 − p+ 3)d2

j ljα2j + 4(d2

j + dj)l2j∂αj

∂lj+ 4d2

j l2jαj

∂αj

∂lj− 4ljαj d

2j(p− j)

+4∑k<j

d2j l

2jαj − d2

kl2kαk

lj − lk+ 4

∑k<j

djljαjlk − dklkαkljlj − lk

]. (35)

Now we observe thatp∑

j=1

(∑k<j

{d2

j l2jαj − d2

kl2kαk

lj − lk+djljαjlk − dklkαklj

lj − lk

}− ljαj d

2j(p − j)

)≤ c(p− p2)

2(b+ u). (36)

Furthermore, noting that d1 < d2 < · · ·< dp and that∑p

j=1 1/{d2j l

2j (b+u)} = u/(b+u) ≤ 1,

we havep∑

j=1

(d2j + dj)l2j

∂αj

∂lj=

c

b+ u

p∑j=1

[−1 +

2d2

j l2j (b + u)

− dj +2dj

d2j l

2j (b+ u)

]

≤ − c

b+ u

[p − 2 − dp +

p−1∑j=1

dj

] ≤ − c

b+ u(p − 2). (37)

The last inequality follows from∑p−1

j=1 dj − dp ≥ 0. Since l1 > l2 > · · · > lp, we have

4p∑

j=1

d2j l

2jαj

∂αj

∂lj=

4c2

(b+ u)2

p∑j=1

[2

d2j l

3j (b+ u)

− 1lj

]≤ 4c2

(b + u)2

[2lp

−p∑

j=1

1lj

],

which givesp∑

j=1

{(n2 − p+ 3)d2

j ljα2j + 4d2

j l2jαj(∂αj/∂lj)

} ≤ c2

(b+ u)2

[8lp

+ (n2 − p − 1)p∑

j=1

1lj

]

≤ (n2 − p+ 7)c2dp

(b+ u)2

p∑j=1

1djlj

≤ (n1 − p− 1)(n2 − p+ 7)c2√p

2(n2 − p+ 3)(b+ u)√b

. (38)

The last inequality follows from the inequality

maxyj>0

∑pj=1 yj

b+∑p

j=1 y2j

≤√p

2√b.

Finally putting (36)–(38) into the rhs of (35), we have

trζ{R2(ζ

BE; ζ) − R2(ζ

LO; ζ)

}≤ EG

θ, Σ

[c

b+ u

{(n1 − p− 1)(n2 − p+ 7)c

√p

2(n2 − p+ 3)√b

− 2(p2 + p− 4)}]

≤ 0.

The last inequality follows from Assumption (III) of Theorem 4. �

–20–


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–22–


Table 1. Estimated risks for estimation of (Σ1, Σ2) under normal distributions

(Estimated standard errors are in parentheses)Σ2Σ

−11 UB ST AI DS AI LO AI

diag(1, 1, 1) 0.9356 0.8938 4.47% 0.6649 28.9% 0.7517 19.7%

(0.0038) (0.0037) (0.0033) (0.0035)

diag(10, 1, 1) 0.9356 0.8938 4.47% 0.7068 24.5% 0.8324 11.0%

(0.0038) (0.0037) (0.0034) (0.0035)

diag(100, 1, 1) 0.9356 0.8938 4.47% 0.7100 24.1% 0.8468 9.48%

(0.0038) (0.0037) (0.0034) (0.0036)

diag(1000, 1, 1) 0.9356 0.8938 4.47% 0.7099 24.1% 0.8480 9.36%

(0.0038) (0.0037) (0.0034) (0.0036)

diag(10, 5, 1) 0.9356 0.8938 4.47% 0.7397 20.9% 0.8331 11.0%

(0.0038) (0.0037) (0.0036) (0.0035)

diag(10, 10, 1) 0.9356 0.8938 4.47% 0.7748 17.2% 0.8325 11.0%

(0.0038) (0.0037) (0.0042) (0.0035)

diag(100, 10, 1) 0.9356 0.8938 4.47% 0.7523 19.6% 0.8812 5.81%

(0.0038) (0.0037) (0.0035) (0.0036)

diag(100, 100,1) 0.9356 0.8938 4.47% 0.8046 14.0% 0.8470 9.47%

(0.0038) (0.0037) (0.0076) (0.0036)

diag(1000, 10, 1) 0.9356 0.8938 4.47% 0.7510 19.7% 0.8879 5.10%

(0.0038) (0.0037) (0.0035) (0.0036)

diag(1000, 100, 1) 0.9356 0.8938 4.47% 0.7694 17.8% 0.8878 5.11%

(0.0038) (0.0037) (0.0035) (0.0036)

diag(1000, 1000, 1) 0.9356 0.8938 4.47% 0.7941 15.1% 0.8482 9.34%

(0.0038) (0.0037) (0.0129) (0.0036)

–23–


Table 2. Estimated risks for estimation of (Σ1, Σ2) under t-distributions



diag(1, 1, 1) 10.116 10.078 0.38% 9.4165 6.92% 9.6676 4.44%

(0.4976) (0.5042) (0.4812) (0.4888)

diag(10, 1, 1) 10.116 10.078 0.38% 9.5315 5.78% 9.8952 2.19%

(0.4976) (0.5042) (0.4838) (0.4937)

diag(100, 1, 1) 10.116 10.078 0.38% 9.5425 5.67% 9.9337 1.81%

(0.4976) (0.5042) (0.4843) (0.4938)

diag(1000, 1, 1) 10.116 10.078 0.38% 9.5423 5.67% 9.9370 1.77%

(0.4976) (0.5042) (0.4843) (0.4937)

diag(10, 5, 1) 10.116 10.078 0.38% 9.6207 4.90% 9.9112 2.03%

(0.4976) (0.5042) (0.4876) (0.4973)

diag(10, 10, 1) 10.116 10.078 0.38% 9.7393 3.73% 9.9116 2.02%

(0.4976) (0.5042) (0.4886) (0.4961)

diag(100, 10, 1) 10.116 10.078 0.38% 9.6664 4.45% 10.040 0.76%

(0.4976) (0.5042) (0.4887) (0.4980)

diag(100, 100,1) 10.116 10.078 0.38% 9.8365 2.77% 9.9501 1.64%

(0.4976) (0.5042) (0.4921) (0.4961)

diag(1000, 10, 1) 10.116 10.078 0.38% 9.6613 4.50% 10.058 0.58%

(0.4976) (0.5042) (0.4887) (0.4980)

diag(1000, 100, 1) 10.116 10.078 0.38% 9.7189 3.93% 10.058 0.58%

(0.4976) (0.5042) (0.4895) (0.4980)

diag(1000, 1000, 1) 10.116 10.078 0.38% 9.8939 2.20% 9.9531 1.61%

(0.4976) (0.5042) (0.5104) (0.4960)

–24–


Table 3. Estimated risks for estimation of (Σ1, Σ2) under Kotz-type distributions



diag(1, 1, 1) 4.6140 4.5725 0.90% 4.5145 2.16% 4.5363 1.68%

(0.0060) (0.0059) (0.0059) (0.0059)

diag(10, 1, 1) 4.6140 4.5725 0.90% 4.5250 1.93% 4.5572 1.23%

(0.0060) (0.0059) (0.0059) (0.0059)

diag(100, 1, 1) 4.6140 4.5725 0.90% 4.5257 1.91% 4.5609 1.15%

(0.0060) (0.0059) (0.0059) (0.0059)

diag(1000, 1, 1) 4.6140 4.5725 0.90% 4.5257 1.91% 4.5612 1.14%

(0.0060) (0.0059) (0.0059) (0.0059)

diag(10, 5, 1) 4.6140 4.5725 0.90% 4.5330 1.76% 4.5572 1.23%

(0.0060) (0.0059) (0.0059) (0.0059)

diag(10, 10, 1) 4.6140 4.5725 0.90% 4.5435 1.53% 4.5570 1.24%

(0.0060) (0.0059) (0.0059) (0.0059)

diag(100, 10, 1) 4.6140 4.5725 0.90% 4.5366 1.68% 4.5696 0.96%

(0.0060) (0.0059) (0.0059) (0.0059)

diag(100, 100,1) 4.6140 4.5725 0.90% 4.5516 1.35% 4.5607 1.16%

(0.0060) (0.0059) (0.0062) (0.0059)

diag(1000, 10, 1) 4.6140 4.5725 0.90% 4.5365 1.68% 4.5713 0.93%

(0.0060) (0.0059) (0.0059) (0.0059)

diag(1000, 100, 1) 4.6140 4.5725 0.90% 4.5408 1.59% 4.5713 0.93%

(0.0060) (0.0059) (0.0059) (0.0059)

diag(1000, 1000, 1) 4.6140 4.5725 0.90% 4.5503 1.38% 4.5610 1.15%

(0.0060) (0.0059) (0.0068) (0.0059)

–25–


Table 4. Estimated risks for estimation of (Σ1, Σ2) under t-distributions (i.i.d.)



diag(1, 1, 1) 8.3045 7.7993 6.08% 7.0405 15.2% 7.2989 12.1%

(0.2110) (0.1879) (0.1845) (0.1849)

diag(10, 1, 1) 8.3045 7.7993 6.08% 7.1560 13.8% 7.5063 9.61%

(0.2110) (0.1879) (0.1847) (0.1853)

diag(100, 1, 1) 8.3045 7.7993 6.08% 7.2055 13.2% 7.6093 8.37%

(0.2110) (0.1879) (0.1853) (0.1861)

diag(1000, 1, 1) 8.3045 7.7993 6.08% 7.2099 13.2% 7.6232 8.20%

(0.2110) (0.1879) (0.1855) (0.1865)

diag(10, 5, 1) 8.3045 7.7993 6.08% 7.1335 14.1% 7.4849 9,87%

(0.2110) (0.1879) (0.1841) (0.1853)

diag(10, 10, 1) 8.3045 7.7993 6.08% 7.2077 13.2% 7.5056 9.62%

(0.2110) (0.1879) (0.1838) (0.1853)

diag(100, 10, 1) 8.3045 7.7993 6.08% 7.2281 13.0% 7.6732 7.60%

(0.2110) (0.1879) (0.1842) (0.1859)

diag(100, 100,1) 8.3045 7.7993 6.08% 7.3523 11.5% 7.6110 8.35%

(0.2110) (0.1879) (0.1852) (0.1862)

diag(1000, 10, 1) 8.3045 7.7993 6.08% 7.2351 12.9% 7.7301 6.92%

(0.2110) (0.1879) (0.1846) (0.1865)

diag(1000, 100, 1) 8.3045 7.7993 6.08% 7.3381 11.6% 7.7286 6.93%

(0.2110) (0.1879) (0.1852) (0.1865)

diag(1000, 1000, 1) 8.3045 7.7993 6.08% 7.3524 11.5% 7.6243 8.19%

(0.2110) (0.1879) (0.1867) (0.1864)

–26–


Table 5. Estimated risks for estimation of (Σ1, Σ2) under Kotz-type distributions (i.i.d.)



diag(1, 1, 1) 1.5235 1.5228 0.04% 1.1793 22.6% 1.3040 14.4%

(0.0037) (0.0037) (0.0030) (0.0033)

diag(10, 1, 1) 1.5235 1.5228 0.04% 1.2461 18.2% 1.4343 5.85%

(0.0037) (0.0037) (0.0032) (0.0035)

diag(100, 1, 1) 1.5235 1.5228 0.04% 1.2489 18.0% 1.4503 4.80%

(0.0037) (0.0037) (0.0032) (0.0036)

diag(1000, 1, 1) 1.5235 1.5228 0.04% 1.2487 18.0% 1.4517 4.71%

(0.0037) (0.0037) (0.0032) (0.0036)

diag(10, 5, 1) 1.5235 1.5228 0.04% 1.3037 14.4% 1.4425 5.32%

(0.0037) (0.0037) (0.0038) (0.0034)

diag(10, 10, 1) 1.5235 1.5228 0.04% 1.3743 9.79% 1.4337 5.89%

(0.0037) (0.0037) (0.0057) (0.0035)

diag(100, 10, 1) 1.5235 1.5228 0.04% 1.3174 13.5% 1.5069 1.09%

(0.0037) (0.0037) (0.0034) (0.0037)

diag(100, 100,1) 1.5235 1.5228 0.04% 1.4237 6.55% 1.4498 4.84%

(0.0037) (0.0037) (0.0136) (0.0036)

diag(1000, 10, 1) 1.5235 1.5228 0.04% 1.3166 13.6% 1.5141 0.62%

(0.0037) (0.0037) (0.0034) (0.0037)

diag(1000, 100, 1) 1.5235 1.5228 0.04% 1.3374 12.2% 1.5141 0.62%

(0.0037) (0.0037) (0.0034) (0.0037)

diag(1000, 1000, 1) 1.5235 1.5228 0.04% 1.4000 8.10% 1.4512 4.74%

(0.0037) (0.0037) (0.0233) (0.0036)

–27–


Table 6. Estimated risks for estimation of Σ2Σ−11 under normal distributions


−11 BU LO AI BE AI SB AI

diag(1, 1, 1) 6.192 5.391 12.9% 4.822 22.1% 4.424 28.6%

(0.033) (0.033) (0.032) (0.032)

diag(10,1, 1) 6.186 6.008 2.87% 5.853 5.38% 5.931 4.12%

(0.042) (0.042) (0.041) (0.042)

diag(100, 1, 1) 6.178 6.160 0.30% 6.142 0.60% 6.151 0.44%

(0.047) (0.047) (0.047) (0.047)

diag(1000, 1, 1) 6.176 6.174 0.03% 6.172 0.06% 6.173 0.05%

(0.047) (0.047) (0.047) (0.047)

diag(10,5, 1) 6.198 5.833 5.89% 5.705 7.96% 5.671 8.67%

(0.037) (0.037) (0.036) (0.036)

diag(10,10, 1) 6.205 5.790 6.69% 5.691 8.28% 5.517 11.1%

(0.037) (0.036) (0.036) (0.036)

diag(100, 10, 1) 6.184 6.124 0.97% 6.106 1.27% 6.167 0.27%

(0.044) (0.044) (0.044) (0.044)

diag(100, 100, 1) 6.205 5.845 5.80% 5.835 5.97% 5.564 10.3%

(0.038) (0.037) (0.037) (0.037)

diag(1000, 10, 1) 6.177 6.171 0.09% 6.169 0.12% 6.176 0.02%

(0.047) (0.047) (0.047) (0.047)

diag(1000, 100,1) 6.183 6.134 0.79% 6.132 0.82% 6.178 0.07%

(0.044) (0.044) (0.044) (0.044)

diag(1000, 1000, 1) 6.204 5.849 5.73% 5.848 5.75% 5.566 10.3%

(0.038) (0.037) (0.037) (0.037)

–28–


Table 7. Estimated risks for estimation of Σ2Σ−11 under t-distributions



diag(1, 1, 1) 10.26 8.954 12.8% 8.033 21.7% 7.359 28.3%

(0.145) (0.131) (0.122) (0.114)

diag(10, 1, 1) 10.14 9.860 2.77% 9.608 5.25% 9.745 3.91%

(0.146) (0.144) (0.141) (0.145)

diag(100, 1, 1) 10.10 10.07 0.29% 10.04 0.58% 10.05 0.44%

(0.152) (0.152) (0.152) (0.152)

diag(1000, 1, 1) 10.09 10.09 0.03% 10.08 0.06% 10.08 0.04%

(0.153) (0.153) (0.153) (0.153)

diag(10, 5, 1) 10.20 9.609 5.83% 9.401 7.86% 9.341 8.46%

(0.143) (0.137) (0.135) (0.134)

diag(10, 10, 1) 10.24 9.568 6.56% 9.408 8.11% 9.123 10.9%

(0.145) (0.139) (0.137) (0.135)

diag(100, 10, 1) 10.12 10.03 0.91% 9.999 1.21% 10.11 0.16%

(0.148) (0.147) (0.147) (0.148)

diag(100, 100, 1) 10.22 9.633 5.71% 9.616 5.87% 9.169 10.3%

(0.145) (0.139) (0.139) (0.135)

diag(1000, 10, 1) 10.09 10.08 0.09% 10.08 0.12% 10.09 0.02%

(0.153) (0.152) (0.152) (0.153)

diag(1000, 100,1) 10.11 10.04 0.74% 10.04 0.77% 10.12 -0.02%

(0.148) (0.147) (0.147) (0.149)

diag(1000, 1000, 1) 10.21 9.631 5.65% 9.630 5.66% 9.165 10.2%

(0.145) (0.139) (0.139) (0.135)

–29–


Table 8. Estimated risks for estimation of Σ2Σ−11 under Kotz-type distributions



diag(1, 1, 1) 1.072 0.933 12.9% 0.835 22.1% 0.766 28.5%

(0.006) (0.006) (0.006) (0.006)

diag(1, 1, 1) 1.074 1.043 2.87% 1.017 5.37% 1.030 4.14%

(0.007) (0.007) (0.007) (0.007)

diag(100, 1, 1) 1.074 1.071 0.30% 1.068 0.59% 1.070 0.44%

(0.008) (0.008) (0.008) (0.008)

diag(1000, 1, 1) 1.074 1.074 0.03% 1.074 0.06% 1.074 0.05%

(0.008) (0.008) (0.008) (0.008)

diag(10,5, 1) 1.073 1.010 5.89% 0.988 7.95% 0.982 8.47%

(0.006) (0.006) (0.006) (0.006)

diag(10,10, 1) 1.072 1.000 6.70% 0.982 8.28% 0.953 11.1%

(0.006) (0.006) (0.006) (0.006)

diag(100, 10, 1) 1.074 1.064 0.96% 1.061 1.25% 1.072 0.25%

(0.008) (0.007) (0.007) (0.008)

diag(100, 100, 1) 1.072 1.009 5.81% 1.008 5.97% 0.961 10.3%

(0.006) (0.006) (0.006) (0.006)

diag(1000, 10, 1) 1.074 1.073 0.09% 1.073 0.12% 1.074 0.01%

(0.008) (0.008) (0.008) (0.008)

diag(1000, 100,1) 1.074 1.066 0.78% 1.065 0.81% 1.074 0.06%

(0.008) (0.008) (0.008) (0.008)

diag(1000, 1000, 1) 1.072 1.010 5.73% 1.010 5.75% 0.961 10.3%

(0.006) (0.006) (0.006) (0.006)

–30–


Table 9. Estimated risks for estimation of Σ2Σ−11 under t-distributions (i.i.d.)



diag(1, 1, 1) 20.73 19.65 5.21% 18.92 8.72% 17.89 13.7%

(0.883) (0.883) (0.888) (0.876)

diag(10,1, 1) 20.48 20.28 0.97% 20.06 2.03% 20.01 2.26%

(0.963) (0.964) (0.965) (0.960)

diag(100, 1, 1) 20.57 20.55 0.06% 20.53 0.18% 20.52 0.20%

(1.140) (1.140) (1.140) (1.140)

diag(1000, 1, 1) 20.63 20.63 0.00% 20.63 0.02% 20.62 0.02%

(1.193) (1.193) (1.193) (1.193)

diag(10,5, 1) 20.14 19.67 2.35% 19.47 3.31% 19.08 5.25%

(0.825) (0.825) (0.826) (0.816)

diag(10,10, 1) 20.02 19.49 2.63% 19.34 3.38% 18.78 6.22%

(0.792) (0.793) (0.793) (0.783)

diag(100, 10, 1) 20.33 20.29 0.20% 20.26 0.34% 20.27 0.27%

(1.008) (1.008) (1.008) (1.006)

diag(100, 100, 1) 19.97 19.52 2.23% 19.50 2.31% 18.85 5.59%

(0.808) (0.809) (0.809) (0.799)

diag(1000, 10, 1) 20.55 20.55 0.00% 20.55 0.01% 20.55 0.01%

(1.149) (1.149) (1.149) (1.149)

diag(1000, 100,1) 20.32 20.29 0.14% 20.29 0.15% 20.29 0.16%

(1.014) (1.014) (1.014) (1.012)

diag(1000, 1000, 1) 19.98 19.53 2.21% 19.53 2.21% 18.87 5.56%

(0.810) (0.810) (0.810) (0.801)

–31–


Table 10. Estimated risks for estimation of Σ2Σ−11 under Kotz-type distributions (i.i.d.)



diag(1, 1, 1) 7.815 6.333 19.0% 5.297 32.2% 4.703 39.8%

(0.029) (0.029) (0.028) (0.029)

diag(10,1, 1) 7.761 7.424 4.35% 7.150 7.88% 7.328 5.58%

(0.042) (0.041) (0.041) (0.043)

diag(100, 1, 1) 7.752 7.714 0.49% 7.682 0.90% 7.703 0.63%

(0.048) (0.048) (0.048) (0.049)

diag(1000, 1, 1) 7.753 7.749 0.05% 7.746 0.09% 7.748 0.06%

(0.049) (0.049) (0.049) (0.049)

diag(10,5, 1) 7.782 7.122 8.49% 6.903 11.3% 7.001 10.0%

(0.035) (0.034) (0.034) (0.033)

diag(10,10, 1) 7.799 7.022 9.96% 6.854 12.1% 6.629 15.0%

(0.034) (0.033) (0.033) (0.034)

diag(100, 10, 1) 7.756 7.633 1.59% 7.602 2.00% 7.731 0.33%

(0.045) (0.045) (0.045) (0.046)

diag(100, 100, 1) 7.796 7.115 8.73% 7.097 8.96% 6.684 14.3%

(0.035) (0.035) (0.035) (0.035)

diag(1000, 10, 1) 7.752 7.740 0.16% 7.736 0.21% 7.750 0.03%

(0.049) (0.049) (0.049) (0.049)

diag(1000, 100,1) 7.756 7.654 1.32% 7.651 1.36% 7.751 0.07%

(0.045) (0.045) (0.045) (0.046)

diag(1000, 1000, 1) 7.795 7.123 8.62% 7.122 8.64% 6.689 14.2%

(0.035) (0.035) (0.035) (0.035)

–32–

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